Let $G$ be a (topologically) simple Hausdorff topological group. Let $H$ be a dense subgroup of $G$. Now throw away the topology. What restrictions are known on the structure of $H$ as an abstract group? I imagine not much can be said if $G$ has a very coarse topology, but I am particularly interested in the case where $G$ is totally disconnected and locally compact, that is, the intersection of all open compact subgroups of $G$ is trivial.

A related question: two (t.d.l.c.) topological groups $G$ and $K$ have a dense subgroup $H$ in common. Suppose $G$ is (topologically) simple. What does this say about $K$?

I don't have a precise question I want to answer here, this is more of an appeal for references on the subject.